The fundamental solution to the Poisson equation with respect to a point-like excitation is known. Such a solution is called Green's function - in three dimensions, we have the famous \(1/r\)-dependency. However, if one knows the Green's function for a differential operator (here Laplace \(\Delta\)) any problem can be solved with respect to the given boundary conditions. In this section we will learn different techniques to solve such boundary value problems, for example the method of mirror charges or direct integrations.
The potential of a dipole in front of a flat metallic surface is calculated. A Taylor expansion is used to understand the resulting electric field.
A derivation of the boundary integral equation in electrostatics. The relation to the Boundary Element Method is explained.
The electrostatic field of a point charge close to a grounded metallic sphere is derived.
A point charge in front of two grounded metallic half-planes, problem and solution.